OFFSET
0,4
LINKS
FORMULA
a(n) = Sum_{k=0..n-1} k*(n-k)!*hypergeom([k-n+1],[2],-k).
a(n) = Sum_{k=0..n-1}(Sum_{j=0.. n-k}((n-k-j)!*C(n-k,j)*C(n-k-1,j-1)*k^j)).
a(n) = Sum_{k=0..n-1} (n-k-1)* k! * LaguerreL(k, 1, k-n+1). - G. C. Greubel, Feb 23 2021
MAPLE
a := n -> add(exp(-k/2)*WhittakerM(-(n-k), 1/2, k)*(n-k)!, k=0..n-1):
seq(round(evalf(a(n), 64)), n=0..22);
# Alternatively:
a := n -> add(k*(n-k)!*hypergeom([k-n+1], [2], -k), k=0..n-1):
seq(simplify(a(n)), n=0..22);
MATHEMATICA
Table[Sum[(n-k-1)*k!*LaguerreL[k, 1, k-n+1], {k, 0, n-1}], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)
PROG
(Sage) [sum( (n-k-1)*factorial(k)*gen_laguerre(k, 1, k-n+1) for k in (0..n-1) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021
(Magma) [n eq 0 select 0 else (&+[(n-k-1)*Factorial(k)*Evaluate( LaguerrePolynomial(k, 1), k-n+1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Mar 24 2015
STATUS
approved